# Properties

 Label 3392.2.a.a Level 3392 Weight 2 Character orbit 3392.a Self dual yes Analytic conductor 27.085 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3392 = 2^{6} \cdot 53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3392.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.0852563656$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 53) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 4q^{7} + 6q^{9} + O(q^{10})$$ $$q - 3q^{3} + 4q^{7} + 6q^{9} + 3q^{13} - 3q^{17} - 5q^{19} - 12q^{21} - 7q^{23} - 5q^{25} - 9q^{27} + 7q^{29} - 4q^{31} - 5q^{37} - 9q^{39} + 6q^{41} - 2q^{43} + 2q^{47} + 9q^{49} + 9q^{51} + q^{53} + 15q^{57} - 2q^{59} + 8q^{61} + 24q^{63} - 12q^{67} + 21q^{69} - q^{71} - 4q^{73} + 15q^{75} + q^{79} + 9q^{81} - q^{83} - 21q^{87} - 14q^{89} + 12q^{91} + 12q^{93} + q^{97} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 0 0 0 4.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3392.2.a.a 1
4.b odd 2 1 3392.2.a.s 1
8.b even 2 1 848.2.a.g 1
8.d odd 2 1 53.2.a.a 1
24.f even 2 1 477.2.a.a 1
24.h odd 2 1 7632.2.a.j 1
40.e odd 2 1 1325.2.a.e 1
40.k even 4 2 1325.2.b.c 2
56.e even 2 1 2597.2.a.a 1
88.g even 2 1 6413.2.a.h 1
104.h odd 2 1 8957.2.a.b 1
424.e odd 2 1 2809.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
53.2.a.a 1 8.d odd 2 1
477.2.a.a 1 24.f even 2 1
848.2.a.g 1 8.b even 2 1
1325.2.a.e 1 40.e odd 2 1
1325.2.b.c 2 40.k even 4 2
2597.2.a.a 1 56.e even 2 1
2809.2.a.a 1 424.e odd 2 1
3392.2.a.a 1 1.a even 1 1 trivial
3392.2.a.s 1 4.b odd 2 1
6413.2.a.h 1 88.g even 2 1
7632.2.a.j 1 24.h odd 2 1
8957.2.a.b 1 104.h odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$53$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3392))$$:

 $$T_{3} + 3$$ $$T_{5}$$ $$T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 + 5 T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - 3 T + 13 T^{2}$$
$17$ $$1 + 3 T + 17 T^{2}$$
$19$ $$1 + 5 T + 19 T^{2}$$
$23$ $$1 + 7 T + 23 T^{2}$$
$29$ $$1 - 7 T + 29 T^{2}$$
$31$ $$1 + 4 T + 31 T^{2}$$
$37$ $$1 + 5 T + 37 T^{2}$$
$41$ $$1 - 6 T + 41 T^{2}$$
$43$ $$1 + 2 T + 43 T^{2}$$
$47$ $$1 - 2 T + 47 T^{2}$$
$53$ $$1 - T$$
$59$ $$1 + 2 T + 59 T^{2}$$
$61$ $$1 - 8 T + 61 T^{2}$$
$67$ $$1 + 12 T + 67 T^{2}$$
$71$ $$1 + T + 71 T^{2}$$
$73$ $$1 + 4 T + 73 T^{2}$$
$79$ $$1 - T + 79 T^{2}$$
$83$ $$1 + T + 83 T^{2}$$
$89$ $$1 + 14 T + 89 T^{2}$$
$97$ $$1 - T + 97 T^{2}$$